Neutrino Mass Matrices with $M_{ee} = 0$

Motivated by the possibility that the amplitude for neutrinoless double beta decay may be much smaller than the planned sensitivity of future experiments, we study ansatze for the neutrino mass matrix with $M_{ee} = 0$. For the case in which CP is conserved, we consider two classes of real-valued mass matrices: "Class I" defined by $|M_{e\mu}| = |M_{e\tau}|$, and "Class II" defined by $|M_{\mu\mu}| = |M_{\tau\tau}|$. The important phenomenological distinction between the two is that Class I permits only "small" values of $V_{e3}$ up to $\sim 0.03$, while Class II admits "large" values of $V_{e3}$ up to its empirical upper limit of 0.22. Then we introduce CP-violating complex phases into the mass matrix. We show that it is possible to have tribimaximal mixing with $M_{ee} = 0$ and $|M_{\mu\tau}| = |M_{\mu\mu}| = |M_{\tau\tau}|$ if the Majorana phase angles are $\pm\pi/4$. Alternatively, for smaller values of $|M_{\mu\tau}| = |M_{\mu\mu}| = |M_{\tau\tau}|$ it is possible to obtain $|V_{e3}| \sim 0.2$ and generate relatively large CP-violating amplitudes. To eliminate phase redundancy, we emphasize rephasing any mass matrix with $M_{ee} = 0$ into a standard form with two complex phases. The discussion alternates between analytical and numerical but remains purely phenomenological, without any attempt to derive mass matrices from a fundamental theory.